Leibniz’s Stepped Reckoner, and a clock for the next 10,000 years

by Charlie Huenemann

In 1671, in some letters exchanged with the French mathematician Pierre de Carcavy, Leibniz mentioned his plans to create a calculating machine. Apparently, he had been inspired by a pedometer, probably thinking that if machines could count, they could then calculate. Within a couple of years, he hired a craftsman build a wooden prototype of his machine, and he packed it along in a trip to London in 1673.

He presented the machine to the Royal Society, but his presentation failed. The machine was supposed to not only add and subtract, but multiply, divide, and even extract square and cube roots. But it just didn't work, though everyone was faintly impressed by the attempt. Well, almost everyone. Robert Hooke made close examination of the machine and asked detailed questions of its inventor. Afterward he set about both disparaging Leibniz's attempt to his friends and making a copy of the machine himself and showing it off.

Leibniz went back home, improved his design, and hired a better craftsman. Then, after two decades of tinkering, adjustments, and debugging, he finally had something to show: the Stepped Reckoner.


A little over a year ago, Stephen Wolfram visited the Leibniz Archive in Hanover and saw the Stepped Reckoner for himself, as well as Leibniz's cramped and ingenious pages of calculations and designs. Wolfram notes that the most interesting “idea before its time” element in the device is Leibniz's appreciation for binary arithmetic. Really, the central idea of modern computing is right there, enshrined in beautifully sculpted gears, with a crank on the side.

Indeed, Leibniz's fascination for binary arithmetic went far beyond its utility for making calculating machines. He found deep metaphysical significance in it. He lived in an age when it was becoming generally accepted that, as Galileo said, nature is a book written in the language of mathematics. And everything in mathematics, Leibniz saw, could be expressed using only 1 and 0 – that is, unity and nothing. And this in turn inspired Leibniz to see God as something like the Divine Coder:

Since all spirits are unities, one can say that God is the primitive unity expressed by all the others according to their capabilities. His goodness moved him to act, and there are in him three primacies, power, knowledge, and will; from this results the operation or the creature, which is varied according to the different combinations of unity with zero, that is of the positive with the privative, since the privative is nothing but the limits and there are limits everywhere in the creature just as there are points everywhere in a line. (Letter to Andreas Morell, 1698; quoted by Antognazza 2011, p. 359)

As Neal Stephenson once wrote: “In the Beginning was the Command Line.”

Leibniz's metaphysical vision, embodied in the Stepped Reckoner, was a heady mix of genuine piety, mechanical philosophy, and a nearly fanatical Pythagoreanism. The necessity found in numbers was beautiful, sacred, and basic to all existence.

Of course, each of us now carries in a pocket a machine vastly more powerful than anything Leibniz designed, but few of us live in awe of a cellphone's metaphysical significance. But something approximating Leibnizian wonder can be evoked by surveying a couple of projects now being carried out by the Long Now Foundation.

The central intent of the Long Now is to get people thinking in longer scales of time. It is pretty well known that many of our economic and geoplolitical problems result from our unwillingness to “jump the gap”, or stay with a project whose range exceeds an election cycle, let alone a generation. But we can change the way we think. Danny Hillis expresses the thought:

When I was a child, people used to talk about what would happen by the year 02000. [Yes; the Long Nowers are anticipating the Y10k problem.] For the next thirty years they kept talking about what would happen by the year 02000, and now no one mentions a future date at all. The future has been shrinking by one year per year for my entire life. I think it is time for us to start a long-term project that gets people thinking past the mental barrier of an ever-shortening future. I would like to propose a large (think Stonehenge) mechanical clock, powered by seasonal temperature changes. It ticks once a year, bongs once a century, and the cuckoo comes out every millennium.

To this end, Hillis and his coworkers have designed and are building the clock. It's powered by differences in daily temperature and by donated labor from curious visitors willing to give the thing a crank. On a smaller scale, Hillis & Co. have built a stunningly beautiful orrery – one of those machines that replicates the planets' orbits around the sun. That's a kind of clock, too – the kind people should be able to figure out even if they are not acquainted with our conventional units of hours and minutes.

In thinking about these projects, the designers had to work themselves backwards from all of the wonders of modern computing technology to the very basics: ones and zeros, expressed with pins pointed up or down and ratchet wheels that have either locked in or haven't. These devices are the first firm embodiments of computational math, at least in our own history of technology.

There are surely other kinds of devices that might also be designed to function for 10,000 years – perhaps some with electronic components and incredible recharging batteries. But my bet is that the designers felt that sort of machine just wasn't as wonderful – at least to us, as we are now. Along with encouraging us to think in larger scales of time, the Long Nowers may also be encouraging us to think in deeper scales of time: that is to say, of the very basic “unity and privation” that underwrites all mechanization.

There is clear value in this deeper way of thinking. Technology is great, but the more advanced it gets, the more likely it is that its fundamental principles will become obscure to us. Without knowing those fundamental principles, we have trouble “unthinking” our way out of technological problems. We become trapped by wrong assumptions about what's essential to a machine (such as that it must be electrical, or must have an hour hand, or a display screen, or something that goes “ping”). We will do more for STEM literacy and innovation in our society not by making sure every kid gets a laptop, but by making sure every kid has a box of gears and axles and some time to play. Hey, it worked for Leibniz.